Fundamental J. Math. Physics, Vol. 1, Issue 1, 2011, Pages 41-52 Published online at http://www.frdint.com/
OCTONION FORMULATION OF SEVEN DIMENSIONAL VECTOR SPACE
BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI
Department of Physics Kumaun University S. S. J. Campus Almora-263601, Uttarakhand India e-mail: [email protected] [email protected]
Abstract
We have generalized the three dimensional space to seven dimensional one by using octonions and their multiplication rules. The identity n (n − 1)(n − 3)(n − 7) = 0 have been verified to show that there exist only n = ,0 1, 3 and 7 dimensions for vector spaces. It is shown that three dimensional vector space may be extended to seven dimensional space by means of octonions under certain permutations of combinations of structure constant associated with the octonion multiplication rules.
According to celebrated Hurwitz theorem there exits [1] four-division algebra consisting of R (real numbers), C (complex numbers), H (quaternions) [2] and O (octonions) [3]. All four algebras are alternative with antisymmetric associators. Real numbers and complex numbers are limited only up to two dimensions, quaternions are extended to four dimensions (one real and three imaginaries) while octonions represent eight dimensions (one scalar and seven vectors, namely, one real and seven imaginaries). There exits a lot of literature [4-12] on the applications of octonions to
Keywords and phras es : three dimensional space, seven dimensional space, octonions, gradient, divergence, curl. Received July 16, 2011 © 2011 Fundamental Research and Development International 42 BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI interpret wave equation, Dirac equation, and the extension of octonion non- associativity to physical theories. Accordingly, in this paper, we have extended the three dimensional vector analysis to seven dimensional one by using octonions and their multiplication rules. Thus, we have redefined the gradient, divergence and curl in seven dimensions by using octonions. The identity n (n − 1) (n − 3) (n − 7) = 0 is verified only for n = ,0 1, 3 and 7 dimensional vectors. It is shown that few vector identities loose their meaning in seven dimensions due to non-associativity of octonions. So, we have reformulated the vector analysis and it is shown that those vector identities loose their original nature are regained under certain permutations of combinations of structure constant associated with the octonion multiplication rules.
• An octonion x is expressed as a set of eight real numbers
7 x = e0 x0 + ∑ e j x j , (1) j =1 where e j ( j = ,1 ,2 ..., 7) are imaginary octonion units and e0 is the multiplicative unit element. Set of octets (e0 , e1, e2 , e3, e4 , e5, e6 , e7 ) is known as the octonion basis elements and satisfies the following multiplication rules
e0 = ;1 e0e j = e je0 = e j ; e jek = −δ jk e0 + f jkl el , ( j, k, l = ,1 ,2 ..., 7). (2)
The structure constant f jkl is completely antisymmetric and takes the value 1 for following combinations
f jkl = + ,1 ∀{( j, k, l) = (123 ), (471 ), (257 ), (165 ), (624 ), (543 ), (736 )}. (3)
It is to be noted that the summation convention is used for repeated indices. Here the octonion algebra O is described over the algebra of real numbers having the vector space of dimension 8. Octonion conjugate is defined as
7 x = e0x0 − ∑ e j x j , (4) j=1 where we have used the conjugates of basis elements as e0 = e0 and eA = −eA. Hence an octonion can be decomposed in terms of its scalar (Sc (x)) and vector (Vec (x)) parts as
OCTONION FORMULATION OF SEVEN DIMENSIONAL … 43
7 1 1 Sc ()x = ()x + x ; Vec ()x = ()x − x = e x . (5) 2 2 ∑ j j j=1
Conjugates of product of two octonions is described as ()xy = y x while the own conjugate of an octonion is written as ()x = x. The scalar product of two octonions is defined as
7 1 1 x, y = ()x y + y x = ()x y + y x = x y . (6) 2 2 ∑ α α α=0
The norm N(x) and inverse x−1 (for a nonzero x) of an octonion are, respectively, defined as
7 x N()x = x x = x x = x2 ⋅ e , x−1 = ⇒ xx −1 = x−1x = .1 (7) ∑ α 0 N()x α=0
The norm N(x) of an octonion x is zero if x = ,0 and is always positive otherwise. It also satisfies the following property of normed algebra
N(xy ) = N(x)N(y) = N(y)N(x). (8)
Equation (2) directly leads to the conclusion that octonions are not associative in nature and thus do not form the group in their usual form. Non-associativity of octonion algebra O is provided by the associator (x, y, z) = (xy )z − x(yz ), ∀x, y, z ∈ O defined for any 3 octonions. If the associator is totally antisymmetric for exchanges of any 2 variables, i.e., (x, y, z) = −(z, y, x) = −(y, x, z) = −(x, z, y), the algebra is called alternative. Hence, the octonion algebra is neither commutative nor associative but is alternative.
• In order to verify the identity n (n − 1) (n − 3) (n − 7) = 0 for n = ,0 1, 3 and 7 dimensional vector spaces, we start with the use of octonion multiplication rules. n = ,0 n = 1 are limited up to scalars while n = 3 dimensional structure is used for three dimensional vector space. Hurwitz theorem states that the vector identities are satisfied for the algebras of real, complex, quaternions and octonions which are, respectively, associated with n = ,0 n = ,1 n = 3 and n = 7 space-dimensions. In order to generalize the three dimensional space to seven dimensional vector space, let
44 BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI us define a vector in seven dimensional vector space over the field of real numbers as A = eˆ1A1 + eˆ2 A2 + eˆ3 A3 + eˆ4 A4 + eˆ5 A5 + eˆ6 A6 + eˆ7 A7 , (9) R where Ai (∀i = ,1 ,2 ,3 ..., 7) ∈ ; and the eˆi ’s (∀i = ,1 ,2 ,3 ..., 7) are described as the octonionic basis elements which satisfy the following vector product rule [3, 12]
k =7 e j × ek = ∑ f jkl el , (∀j, k, l = ,1 ,2 ..., ,6 7), (10) k =1 where f jkl is the structure constant (3) which is a totally G2 - invariant anti- symmetric tensor. As such, we have
f jkl flmn = g jkmn + δ jm δkn − δ jn δkm = − flmn fkjl , (11) where g jkmn = e j ⋅ {ek , em , en } is a totally G2 - invariant anti-symmetric tensor
[11] and g jkmn = −gmkjn . The only independent components are g1254 = g1267 = g1364 = g1375 = g2347 = g2365 = g4576 = .1
• Hence, the following vector cross products are verified in seven dimensional, i.e., A × A = ,0 ()()A × B ⋅ B = A × B ⋅ A = ,0 A × B = A B , {(}∀ A ⋅ B = 0 , ()()()()()A × B ⋅ A × B = A ⋅ A B ⋅ B − A ⋅ B 2 , A × ()()()B × A = A ⋅ A B − A ⋅ B A. (12) • Here, we observe that the vector triple cross product identity A × ()B × C = B()()A ⋅ C − C A ⋅ B of three dimensional vector space is not satisfied in seven dimensional vector space. So, we may write the left hand side of A × ()()()B × C = B A ⋅ C − C A ⋅ B as
7 7 A × ()B × C = −∑ ∑ []g pqjk + δ pj δqk − δ pk δqj Aj BpCqeˆk jk=1 pq =1
OCTONION FORMULATION OF SEVEN DIMENSIONAL … 45
77 7 7 = −∑∑ g pqjk Aj B pCqeˆk − ∑Aj B jCk eˆk + ∑ AjC j Bk eˆk jkpq jk jk = irreducibl e term + B()()A ⋅ C − C A ⋅ B . (13)
This identity of vector triple product in seven dimensional vector space is satisfied under the octonion multiplication rule by taking the irreducible term as a ternary product such that
7 7 irreducibl e term = −∑∑ g pqjk Aj BpCqeˆk = {}A, B, C , (14) jk pq which gives rise to A × ()()(){}B × C = B A ⋅ C − C A ⋅ B + A, B, C . (15) The {}A, B, C is now be re-called as the associator for octonions. Under the condition of alternativity of octonions, equation (15) reduces to the well-known vector identity A × ()()()B × C = B A ⋅ C − C A ⋅ B for the various permutation values of structure constant f jkl . As such, the vector product in seven dimensional vector space is verified and the mutual rotation of three different vectors in seven dimensional space is possible only for the certain permutations of combination of basis vectors.
• Let us check the following identity of three dimensional vector space ()()A × B × C × D = [ A ⋅ ()B × D ][C − A ⋅ ()B × C ]D (16) to our seven dimensional case. The left hand side of equation (16) now reduces to
7 7 7 ()()A × B × C × D = f f A B f C D eˆ ∑knp ∑ ijk i j ∑ lmn l m p knp ijk lmn
7 7 = ˆ ∑ ()fknp flmn Cl Dm ep ∑ fijk Ai B j lmknp ijk
7 7 = ∑ ()()fijm Ai B j Dm C peˆ p − ∑ fijl Ai B jCl Dpeˆp ijmp ijlp
46 BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI
7 − ∑ glmkp Cl Dm ()fijk Ai B j eˆ p (17) lmkp while the right hand is to be written as [ A ⋅ ()B × D ][C − A ⋅ ()B × C ]D
7 7 = ∑ ()()f xyz Az Bx Dy Cteˆt − ∑ fegh Ah BeCg Dveˆv. (18) xyzt eghv
Comparing equations (17) and (18), we get ()()A × B × C × D = [ A ⋅ ()B × D ][C − A ⋅ ()B × C ]D + irreducibl e term . (19)
The irreducible term is not anything but the octonion associator as
7 irreducibl e term = −∑ glmkp Cl Dm (){}fijk Ai B j eˆ p = A × B, C, D . (20) lmkp
Thus, the seven dimensional analogue of equation (16) is described as ()()A × B × C × D = [ A ⋅ ()B × D ][C − A ⋅ ()B × C ]D + {}A × B, C, D , (21) where the associator {}A × B, C, D reduces to be vanishing under the octonion multiplication rule. Hence the vector-identity (16) is verified for seven dimensions. It means that the three dimensional space can be generalized to seven dimensional space only if certain combinations of basis vectors are allowed with the possible permutations of structure constant (3).
• Let us verify the another identity of three dimensional vector space, i.e., ()()()()()()A × B ⋅ C × D = A ⋅ C B ⋅ D − A ⋅ D B ⋅ C . (22)
The left hand side of this equation reduces to 7 ()()A × B ⋅ C × D = ∑ fijk flmk Ai B jCl Dm ijlm
7 7 7 = ∑ AiCi B j D j − ∑Ai Di B jC j − ∑ gijlm Ai B jCl Dm ij ij ijlm
OCTONION FORMULATION OF SEVEN DIMENSIONAL … 47
7 = −∑ gijlm Ai B jCl Dm + ()()()()A ⋅ C B ⋅ D − A ⋅ D B ⋅ C ijlm = irreducibl e term + ()()()()A ⋅ C B ⋅ D − A ⋅ D B ⋅ C , (23) where the irreducible term can be simplified and accordingly we may write it as the scalar product of a ternary product of three vectors, i.e., 7 irreducibl e term = −∑ gijlm Ai B jCl Dm = {}A, B, C ⋅ D. (24) ijlm
So, equation (23) changes to ()()()()()(){}A × B ⋅ C × D = A ⋅ C B ⋅ D − A ⋅ D B ⋅ C + A, B, C ⋅ D, (25) where the irreducible term may be considered vanishing for the octonion associator for the allowed permutation of structure constants. Hence, the vector identity equation (22) is verified for seven dimensional vector space.
• Similarly, we can verify the following vector identities in seven dimensional space time, i.e., A × ()()()B × C + B × C × A + C × A × B = 0 (26) and A × [B × ()C × D ] = ()()()()A × C B ⋅ D − A × D B ⋅ C , (27) which are also satisfied in seven dimensional space for certain combinations of permutations of structure constant.
• Accordingly, we may develop the vector calculus for seven dimensional vector space in terms of octonion variables. Let us define the seven dimensional differential 7 ∂ operator Nabla as ∇ = ∑ eˆi . So, the gradient, curl and divergence of scalar i ∂xi and vector quantities are described as
7 ∂u grad u = ∇u = eˆ , (28) ∑ i ∂x i i 7 ∂Aj curl A = ∇ × A = f eˆ , (29) ∑ ijk ∂x k ijk i
48 BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI
7 ∂A div A = ∇ ⋅ A = i . (30) ∑ ∂x i i
• The Divergence of a curl is shown to be vanishing as
7 7 7 7 2 ∂ ∂Aj ∂ Aj ∇ ⋅ ()∇ × A = eˆ ⋅ f eˆ = f δ ∑ l ∂x ∑ ijk ∂x k ∑∑ ijk ∂x ∂x lk l l ijk i l ijk l i
72 7 2 ∂ A j 1 ∂ Aj = f = []f + f = .0 (31) ∑ijk ∂x ∂x 2 ∑ ijk kji ∂x ∂x ijkk i ijk i k
Hence curl of a vector is also solenoidal in seven dimensional vector space.
• We may now write the curl of a gradient in 7-dimensional vector-space by adopting the octonion multiplication rules (2) as
7 7 ∂ ∂u ∇ × ()∇u = f eˆ ∑ ijk ∂x ∑ ∂x k ijk i j j
7 7 ∂2u 1 ∂2u = f eˆ = []f + f eˆ = .0 (32) ∑ ijk ∂x ∂x k 2 ∑ ijk jik ∂x ∂x k ijk i j ijk j i
Thus, the gradient of a curl is also irrotational in seven dimensional vector space under octonion multiplication rules.
• Let us check the following vector identity of three dimensions ∇ ⋅ ()()()A × B = B ⋅ ∇ × A − A ⋅ ∇ × B (33) in seven dimensional vector-space. The left hand side of equation (33) reduces to
7 7 ∂ ∇ ⋅ ()A × B = eˆ ⋅ f A B eˆ ∑l ∂x ∑ ijk i j k ll ijk
7 7 7 ∂ ∂B j ∂A = f []A B δ = f A + B i (34) ∑∑ijk ∂x i j lk ∑ ijk i ∂x j ∂x ijkll ijk k k while the right hand side of equation (33) changes to
OCTONION FORMULATION OF SEVEN DIMENSIONAL … 49 B ⋅ ()()∇ × A − A ⋅ ∇ × B
7 7 7 7 ∂ ∂An Bq = eˆl Bl ⋅ fmnt eˆt − eˆs As ⋅ f pqr eˆr ∑ ∑ ∂x ∑ ∑ ∂x l mnt m s pqr p
7 7 ∂ 7 ∂ ∂Ai B j B j ∂Ai = f B − − f A = f A + B , (35) ∑ijk j ∂x ∑ ijk i ∂x ∑ ijk i ∂x j ∂x ijkk ijk k ijk k k which is equal to the left hand side of equation (34). Hence the identity (33) is satisfied for seven dimensional space.
• Similarly, on using the octonion multiplication rules (2), we may prove the following identities in seven dimensions, i.e., ∇ ⋅ ()()()uA = u ∇ ⋅ A + A ⋅ ∇u , (36) ∇ × ()()()uA = u ∇ × A + ∇u × A, (37) ∇ × ()()∇ × A = ∇ ∇ ⋅ A − ∇2 A. (38)
• However the vector identity of three dimensional vector space ∇ × ()()()()()A × B = B ⋅ ∇ A − A ⋅ ∇ B − B ∇ ⋅ A + A ∇ ⋅ B (39) is not satisfied in seven dimension and its left hand side reduces to
7 7 ∂ ∇ × ()A × B = f f A B eˆ ∑ wkt ∂x ∑ ijk i j t wkt w ijk
7 7 ∂ = − f f []A B eˆ (40) ∑∑ ijk kwt ∂x i j t wkt ijk w whereas the right hand expression of equation (39) becomes ()()()()B ⋅ ∇ A − A ⋅ ∇ B − B ∇ ⋅ A + A ∇ ⋅ B
7 7 7 7 ∂A ∂B ∂A ∂B j = i B eˆ − A m eˆ − i B eˆ + A eˆ . (41) ∑∂x j i ∑ l ∂x m ∑∂x j j ∑ i ∂x i ijj lm l iji ij j
50 BHUPENDRA C. S. CHAUHAN and O. P. S. NEGI
Comparing equations (40) and (41), we get ∇ × ()()()()()A × B = B ⋅ ∇ A − A ⋅ ∇ B − B ∇ ⋅ A + A ∇ ⋅ B + irreducibl e term , (42) where the irreducible term takes the expression
7 7 ∂ irreducibl e term = − g []A B eˆ ∑∑ ijwt ∂x i j t wt ij w
7 7 ∂ = − g []A B eˆ ∑∑ ijwt ∂x i j t wt ij w
7 = ∑ eˆt [∇ ⋅ {}eˆt , A, B ]. (43) t
Hence, we get
7 ∇ × ()A × B + ∑ eˆt [∇ ⋅ {}eˆt , A, B ] t = ()()()()B ⋅ ∇ A − A ⋅ ∇ B − B ∇ ⋅ A + A ∇ ⋅ B . (44)
Thus we see that the identity (39) is satisfied only when we have the reducible term vanishing and that can be obtained for different values of permutations of structure constant given by equation (3).
• Similarly, the other vector identity A × ()()()()()∇ × B + B × ∇ × A + A ⋅ ∇ B + B ∇ ⋅ A = ∇ A ⋅ B (45) is not satisfied in general in seven dimensional and takes the following expression A × ()()()()∇ × B + A ⋅ ∇ B + Bˆ × ∇ × A + B ⋅ ∇ A = ∇()A ⋅ B + irreducibl e term , (46) where the irreducible term is described as
7 7 ∂Bt ∂At irreducibl e term = − gstik Ai eˆk + gstik Bi eˆk ∑∂x ∑ ∂x iksts ikst s
OCTONION FORMULATION OF SEVEN DIMENSIONAL … 51
7 7 = eˆk [ A ⋅ {}eˆk , ∇, B ][+ eˆk B ⋅ {}eˆk , ∇, A ]. (47) ∑ ∑ k k
So using the alternativity of octonions for allowed permutations of octonions, we get A × ()()()()∇ × B + B × ∇ × A + A ⋅ ∇ B + B ∇ ⋅ A
7 7 = ∇()A ⋅ B − ∑ eˆk [ A ⋅ {}eˆk , ∇, B ][+ ∑ eˆk B ⋅ {}eˆk , ∇, A ], (48) k k where the irreducible term has been terminated for the different allowed permutations of structure constant discussed above.
From the foregoing analysis, we may conclude that the generalization of three dimensional vector space to seven dimensional vector space needs some modifications. The scalar product of three different vectors retain their form in seven dimensional space but the vector triple product of three different vectors looses its usual form of three dimensional space in seven dimensional vector space. It regains its usual form only for certain combination of allowed value of permutations of structure constant fijk of octonion units. It means when three different vectors are mutually rotated in seven dimensional space their rotation is only possible in certain directions as allowed by the combinations of basis vectors. Similarly, the mutual rotation of four vectors also allowed for certain combination of basis vectors. Consequently, the physical variables (scalar or vector) which are solenoidal and irrotational in three dimensional vector space are also solenoidal and irrotational in seven dimensional vector space. On the other hand, there are vector identities which changes their forms in seven dimensional space and regain their usual form under allowed permutations of structure constant.
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